Tangent spaces and derivatives

Partial derivatives with respect to local coordinates

Let U\to\mathbf{R}^n be a smooth chart defined over an open set U in M. Then there are smooth real-valued functions x^1,x^2,\dots,x^n on U such that

\displaystyle \varphi(u)=(x^1(u),x^2(u),\dots,x^n(u))

for all u\in U. The functions x^1,x^2,\dots,x^n determined by the chart (U,\varphi) constitute smooth local coordinate functions defined over the domain U of the chart.

Definition Let x^1,x^2,\dots,x^n be smooth local coordinates defined over an open set U in a smooth manifold M of dimension n, and let V be the corresponding open set in \mathbf{R}^n defined such that

 

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